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Springer

Received: June Revised: July Accepted: August Published: September

10, 22, 12, 11,

2020 2020 2020 2020

Xinyi Chen-Lin and Amit Dekel Nordita, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden

E-mail: [email protected], [email protected] Abstract: A new supersymmetric D7-brane embedding in the Pilch-Warner gravitational background is found exactly, by solving the supersymmetric condition. In the dual holographic picture, our setting corresponds to adding a quenched fundamental matter sector to N = 2∗ super Yang-Mills theory, at zero temperature. We show that previous results in the same setting are missing the Wess-Zumino term in the D-brane action, and how our results complete the picture. Keywords: D-branes, AdS-CFT Correspondence, Gauge-gravity correspondence ArXiv ePrint: 2006.03024

c The Authors. Open Access, Article funded by SCOAP3 .

https://doi.org/10.1007/JHEP09(2020)087

JHEP09(2020)087

Exact D7-brane embedding in the Pilch-Warner background

Contents 1

2 D-branes 2.1 Kappa symmetry projector 2.2 Supersymmetric condition 2.3 Action

2 2 3 4

3 D7-brane 3.1 Kappa symmetry projector 3.2 Supersymmetric condition 3.3 Solution

4 5 5 6

4 Action 4.1 Wess-Zumino term 4.2 Equation of motion 4.3 Holographic renormalization 4.3.1 Large L

7 8 8 8 10

5 Conclusion

11

A Pilch-Warner supergravity A.1 Metric A.2 Local frame A.3 The near-boundary geometry A.4 Background fields A.5 Killing spinor

12 12 13 14 14 15

B Integrate on-shell Lagrangian B.1 UV limit B.2 IR limit and large L

16 17 17

1

Introduction

The holographic principle promises to be a useful framework to tackle strongly coupled gauge theories by means of weakly coupled string theories. Its best known instance is the AdS/CFT duality, conceived by Maldacena in his seminar work [1]. In particular, the duality involves a strongly coupled SU(N ) N = 4 SYM, which is a CFT, and the supergravity in the AdS5 × S 5 background. This is a very well understood case by now,

–1–

JHEP09(2020)087

1 Introduction

2 2.1

D-branes Kappa symmetry projector

For any D-brane configuration there is an associated kappa symmetry projector, which, in Minkowski signature, is given by [12]: eF ∧ X|Vol dp+1 ξ Γ = − p , − det(g + F )

–2–

(2.1)

JHEP09(2020)087

and despite not describing any real world systems, AdS/CFT is being used as a theoretical laboratory to explore dynamics of real strongly coupled systems. The most famous example is undoubtedly Quantum Chromodynamics (QCD). To compare N = 4 SYM with QCD, we must first add flavors to AdS/CFT. This means additional hypermultiplet sectors with fields in the fundamental representation of the gauge group SU(N ). The equivalent in the holographic picture is the insertion of probe D7-brane embeddings in AdS5 × S 5 , see [2]. The fundamental matter fields, or quarks, arise from strings stretched between the D7 and the N coincident D3-branes that generate the supergravity background. When the D7-branes and the D3-branes are separated, the quarks become massive. Significant work has been done in thi